Monty Hall problem
__TOC__ The Problem : You are a contestant on a game show hosted by Monty Hall. Near Monty, on the stage, are three curtains. Everything Monty will tell you about these curtains is the truth. :* Monty explains that behind one of the curtains there is a luxury automobile, the grand prize. A goat is behind each of the other two curtains. :* Monty explains further that he, himself, already knows which curtain hides the automobile, but of course he will not tell you. You have to guess. :* If you can guess which curtain hides the automobile, Monty says, you will win the grand prize. If, however, you select a curtain that hides a goat, you will win only a consolation prize, the "Let's Make a Deal" Board Game. It's not much, but it's better than nothing. :* So you seem to have a 1/3 chance of winning the grand prize. But there's a twist. Monty next explains that after you make your initial selection, he will open a curtain and reveal to you a goat behind one of the two doors that you did not select. He will then say, "Let's make a deal!" :* The deal Monty says he will offer is that, before any other curtain is opened, you may change your selection. :* If you stick with your original choice, Monty tells you, you are guaranteed to at least go home with the board game. But if you switch your choice to the other closed curtain and lose, you will forfeit the consolation prize and walk away with nothing. : Monty asks if you understand what is about to happen. You say you do. The game begins. You make your curtain choice and you announce it to Monty. He then walks over to one of the other two curtains and opens it, revealing a goat exactly as he had promised. And then he offers you the deal he promised: stick or switch. : The Monty Hall problem, then, is simply this: Should you: :: (A) stick with your original choice because you believe the odds are not improved by switching, and so you might as well go home with at least the consolation prize; or, :: (B) switch to the other closed curtain because you believe this substantially increases your chance of winning the grand prize? The Solution : Common sense seems to tell us that the chance of winning cannot be improved by switching. After all, there are just two closed curtains left, and the automobile is behind one of them. If you stick, or switch, you should have a 1/2 chance of winning. : But common sense is not always reliable. Think of it this way: Suppose you decide to announce, not only your initial choice, but also your "stick or switch" decision at the beginning, just after Monty has explained the game. : Following this procedure, which you are entirely free to adopt, you would tell Monty that you understand the game and then you would announce: :: (A) "Monty, I choose the blue curtain, and I will ' stick ' with this choice after you reveal the goat;" or, :: (B) "Monty, I choose the blue curtain, and I will switch from this choice after you reveal the goat." : Which would you announce? If you choose (A) stick, you know that you will win the automobile only if it is behind the curtain you chose, a 1/3 chance of winning. But if you choose (B) switch, you know that you will win the automobile if it is behind either of the other two curtains, a 2/3 chance of winning! : The correct answer is, therefore, you should switch. The Controversy : The solution to this puzzle ignited a storm of controversy when it appeared in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990. Marilyn explained the answer carefully, but many mathematicians, educators and scientists wrote letters disagreeing with and even ridiculing her. Finally, she challenged schoolchildren to perform an experiment by playing the game hundreds of times and tabulating the results. The response was tremendous and showed the doubters to be wrong and Marilyn right. : A small minority of doubters pointed out that Marilyn's description of the puzzle did not make clear that Monty would always offer the deal no matter which curtain was chosen. If Monty wanted the contestant to lose, they suggested, then he might offer the deal only if the contestant chose the curtain hiding the grand prize. To these critics, she responded that such an interpretation was not what she meant, and was "a different question." A Different Question : Here's another different question: Suppose that Monty did not know which curtain hides the automobile. : In this new scenario, Monty explains that he himself does not know where the automobile is. After you make your initial choice he will open one of the curtains that you did not choose. :* If the curtain Monty opens hid the automobile, you will lose but you will go away with your consolation prize. :* If, however, the curtain hid a goat, you will then have the opportunity to switch from your initial choice, but you will forfeit the consolation prize if you do so. : Let's say that Monty has opened a curtain and a goat is revealed. Now, should you switch from or stick to the choice you made? : The problem at this point looks identical to the original. You've chosen a curtain, and Monty has revealed a goat behind one of the other curtains. Since switching gave you a 2/3 chance of winning in the original problem, common sense seems to say, now, that you ought to have the exact same opportunity in this one. : But again, common sense is not always reliable. : Following the advance announcement procedure, which you are still entirely free to adopt, you would choose between: :: (A) "Monty, I choose the blue curtain, and I will ' stick ' with this choice if you reveal a goat;" or, :: (B) "Monty, I choose the blue curtain, and I will switch from this choice if you reveal a goat." : In this case, the (A) stick decision still wins only if the automobile is behind the curtain you chose, a 1/3 chance. However, the (B) switch decision does not win if the automobile is behind either of the other curtains. It only wins if the automobile is behind one of them, the one that Monty does not choose. So, selecting the switch option in this case does nothing to improve your chances. You should stick, to retain the consolation prize. Summing it up : If the above is not perfectly clear, imagine that you play the game 300 times. Each time the automobile is randomly put behind one of the curtains, and each time your decision will be to switch when you have the opportunity to do so. :* In the original scenario, Monty knows where the automobile is and will always reveal a goat behind a curtain you did not choose. In the second scenario, he doesn't know, and so he will sometimes reveal the automobile, in which case you lose. :* So, under either scenario, you lose when you initially select the automobile's curtain, about one-third or 100 times. (If you've selected the automobile, Monty will reveal a goat in either scenario, and so you'll switch to a losing curtain.) First or second scenario: 100 losses (the number of times you initially selected the automobile). :* But what about the other 200 times (when you've initially selected a curtain that hides a goat)? In the first scenario you always win because Monty always reveals a goat. But in the second scenario, you lose the 100 times that Monty happens to reveal the automobile, and you win only the other 100 times. First scenario: 200 wins. Second scenario: 100 wins. : Or, we might simply list all the possible outcomes of a switch decision in each scenario. In the first, Monty will always reveal a goat no matter what curtain you select. In the second, he'll reveal a goat if you initially select the auto, but if you select a goat, he'll reveal either a goat or an auto. So the complete list of outcomes looks like this: : : By switching you win 2 out of 3 times in the first scenario, and 1 out of 3 times in the second.